Alzheimer’s disease (AD) is a neurodegenerative disorder which has become an outstanding social problem. The main objective of this study was to evaluate the alterations that dementia due to AD elicits in the distribution of functional network weights. Functional connectivity networks were obtained using the orthogonalized Amplitude Envelope Correlation (AEC), computed from source-reconstructed resting-state eletroencephalographic (EEG) data in a population formed by 45 cognitive healthy elderly controls, 69 mild cognitive impaired (MCI) patients and 81 AD patients. Our results indicated that AD induces a progressive alteration of network weights distribution; specifically, the Shannon entropy (SE) of the weights distribution showed statistically significant between-group differences (p < 0.05, Kruskal-Wallis test, False Discovery Rate corrected). Furthermore, an in-depth analysis of network weights distributions was performed in delta, alpha, and beta-1 frequency bands to discriminate the weight ranges showing statistical differences in SE. Our results showed that lower and higher weights were more affected by the disease, whereas mid-range connections remained unchanged. These findings support the importance of performing detailed analyses of the network weights distribution to further understand the impact of AD progression on functional brain activity.